Translational order

Dislocations destroy translational order (shearing along the red arrows), but orientational order is still visible, indicated by black lines in one direction (upper part). Disclinations additionally destroy orientational order (lower part).

If the position of atoms or particles is known, then the translational order can be determined with the translational correlation function ${\displaystyle G_{\vec {G}}({\vec {R}})}$ as function of the distance between lattice site at place ${\displaystyle {\vec {R}}}$ and the place ${\displaystyle {\vec {0}}}$, based on the two-dimensional density function ${\displaystyle \rho _{\vec {G}}({\vec {R}})=e^{i{\vec {G}}\cdot [{\vec {R}}+{\vec {u}}({\vec {R}})]}}$ in reciprocal space:

${\displaystyle G_{\vec {G}}({\vec {R}})=\langle \rho _{\vec {G}}({\vec {R}})\cdot \rho _{\vec {G}}^{\ast }({\vec {0}})\rangle }$

The vector ${\displaystyle {\vec {R}}}$ points to a lattice site within the crystal, where the atom is allowed to fluctuate with an amplitude ${\displaystyle {\vec {u}}({\vec {R}})}$ by thermal motion. ${\displaystyle {\vec {G}}}$ is a reciprocal vector in Fourier space. The brackets denote a statistical average about all pairs of atoms with distance R an.

The translational correlation function decays fast, i. e. exponential, in the hexatic phase. In a 2D crystal, the translational order is quasi-long range and the correlation function decays rather slow, i. e. algebraic; It is not perfect long range, as in three dimensions, since the displacements ${\displaystyle {\vec {u}}({\vec {R}})}$ diverge logarithmically with systems size at temperatures above T=0 due to the Mermin-Wagner theorem.

A disadvantage of the translational correlation function is, that it is strictly spoken only well defined within the crystal. In the isotropic fluid, at the latest, disclinations are present and the reciprocal lattice vector is not defined any more.

Orientational order

The j nearest neighbours of the i-th particle define the sixfold director field.

The orientational order can be determined by the local director field of a particle at place ${\displaystyle {\vec {r}}_{i}}$, if the angles ${\displaystyle \theta _{ij}}$ are taken, given by the bond to the ${\displaystyle N_{i}}$ nearest neighbours in sixfolded space, normalized with the number of nearest neighbours:

${\displaystyle \Psi ({\vec {r}}_{i})={\frac {1}{N_{i}}}\sum _{j=1}^{N_{i}}e^{i6\theta _{ij}}}$

${\displaystyle \Psi }$ is a complex number of magnitude ${\displaystyle |\Psi ({\vec {r}})|\leq 1}$ and the orientation of the six-folded director is given by the phase. In a hexagonal crystal, this is nothing else but the crystal-axes. The local director field disappears for a particle with five or seven nearest neighbours, as given by dislocations and disclinations ${\displaystyle \Psi \sim 0}$, except a small contribution due to thermal motion. The orientational correlation function between two particles i and k at distance ${\displaystyle {\vec {r}}={\vec {r}}_{i}-{\vec {r}}_{k}}$ is now defined using the local director field:

${\displaystyle G_{6}({\vec {r}})=\langle \Psi ({\vec {r}}_{i})\cdot \Psi ^{\ast }({\vec {r}}_{k})\rangle }$

Again, the brackets denote the statistical average about all pairs of particles with distance ${\displaystyle |{\vec {r}}|=r}$. All three thermodynamic phases can be identified with this orientational correlation function: it does not decay in the 2D crystal but takes a constant value (shown in blue in the figure). The stiffness against local torsion is arbitrarily large, Franks's constant is infinity. In the hexatic phase, the correlation decays with a power law (algebraic). This gives straight lines in a log-log-plot, shown in green in the Figure. In the isotropic phase, the correlations decay exponentially fast, this are the red curved lines in the log-log-plot (in a lin-log-plot, it would be straight lines). The discrete structure of the atoms or particles superimposes the correlation function, given by the minima at half integral distances ${\displaystyle a}$. Particles which are poorly correlated in position, are also poorly correlated in their director.

Orientational correlation of the local director field as function of distance, plotted in blue for the crystal, green for the hexatic phase and red for the isotropic fluid.